Question

Prove that √2 + √5 is an irrational number.
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Answer 1

Hey mate..here is yur answer...

If possible,let root 2 + root 5 be rational..

Then,

root 2 + root 5 is rational..

=> root 2 + root 5 whole square is rational..

=> 7 + 2 root 10

Thus we arrive at a contradiction

This contradiction arrives as we took root 2 + root 5 rational..

Therefore,root 2 + root 5 is an irrational number..

Hope this helped...

Answer 2

Let √2 + √5 be a rational number.

Rational numbers can be expressed in the form a/b where a and b are co-prime and b ≠ 0

$\sqrt{2}+\sqrt{5}=\dfrac{a}{b}$

Squaring both sides, we get,

$\implies (\sqrt{2}+\sqrt{5})^{2} =\dfrac{a^{2} }{b^{2} }$

$\implies 2+2\sqrt{10}+5=\dfrac{a^{2} }{b^{2} }$

$\implies 7+2\sqrt{10}=\dfrac{a^{2} }{b^{2} }$

$\implies 2\sqrt{10}=\dfrac{a^{2} }{b^{2} } - 7$

$\implies 2\sqrt{10}=\dfrac{a^{2}-7b^{2} }{b^{2} }$

$\implies \sqrt{10}=\dfrac{a^{2}-7b^{2} }{2b^{2} }$

RHS is a rational number.

=> √10 is a rational number.

But this contradicts to the fact that it a rational number.

Hence, our assumption is wrong.

$\boxed{\boxed{\bold{Therefore, \ \sqrt{2} +\sqrt{5} \ is \ an \ irrational \ number.}}}}}}$